Turning control of a 3- Joint carangiform fish robot using sliding mode based controllers

In this paper, propose an efficient advanced controller that runs well in controlling the motion for fish robot. The fish robot is a new type of biomimetic underwater robot which is developing very fast in recent years by many researchers.

Trang 14

Turning control of a 3- Joint carangiform fish robot using sliding mode based controllers

 Tuong Quan Vo

University of Technology, VNU – HCM

ABSTRACT:

The fish robot is a new type of

biomimetic underwater robot which is

developing very fast in recent years by

many researchers Because it moves

silently, saves energy, and is flexible in

its operation in comparison to other

kinds of underwater robots, such as

Remotely Operated Vehicles (ROVs) or

(AUVs) In this paper, we propose an

efficient advanced controller that runs

well in controlling the motion for our fish

robot First, we derive a new dynamic

model of a 3-joint (4 links) Carangiform fish robot The dynamic model also addresses the heading angle of a fish robot, which is not often covered in other research Second, we present a Sliding Mode Controller (SMC) and a Fuzzy Sliding Mode Controller (FSMC) to the straight motion and turning motion of a fish robot Then, in order to prove the effectiveness of the SMC and FSMC, we conduct some numerical simulations to show the feasibility or the advantage of these proposed controllers

Keywords: Dynamic modeling, Fish robot, Straight, Turning, Sliding mode controller,

SMC, Fuzzy sliding mode controller, FSMC

1 INTRODUCTION

propulsion mainly depend on the use of

propellers or thrusters to generate the motion for

objects in underwater environments However,

most marine animals use the undulation of their

body shape, as well as oscillation of their tail

fins, to generate propulsive force The changing

of body shape generates propulsion to make the

object move forward or backward effectively

The Carangiform-type fish is a kind of changing

body shape that creates motion in the underwater

environment

George V Lauder and Eliot G Drucker

thoroughly surveyed and analyzed the motion

mechanisms of fish fins in order to develop such

a successful underwater robot system [1] M J

Lighthill also surveyed the hydromechanics of

aquatic animal propulsion because of many kinds

generations to adapt to the harsh underwater environment [2] Iman Borazjani and Fotis

Carangiform swimming in the transitional and flow regimes [3] They employed numerical simulation to investigate the hydrodynamics of Carangiform locomotion, including the relative magnitudes of the viscous and inertial forces, i.e the Reynolds number (Re) and the tail-beat frequency or Strouhal number (St), which were systematically varied [3] Guo Jenhwa developed

a measurement strategy for a biomimetic autonomous underwater vehicle (BAUV) in order

to reduce positioning uncertainties while the BAUV was controlled to reach a target efficiently [4] The BAUV plays the role of a

target tracker and can swing its pectoral fins to

adjust its direction when searching for a target;

the BAUV oscillates its tail fin to move forward

to the target [4] Also, K H Low et al.[5]

discussed design mechanisms, the planar serial

chain mechanism, and the parallel mechanism in

their report of a gait study of biomimetic fish

robots using either mechanism In addition, the

authors also discussed the gait functions for the

two forms of biomimetic fish robots [5]

Other research focuses on the heading control

problem of fish robots or related underwater

robot types Some of the intelligent controllers

were proposed by many researchers Jenhwa Guo

used Genetic Algorithms to find the body spline

parameter values of a fish robot He then

developed a control law that satisfies the

Lyapunov function in the heading control of a

BAUV [6] J Guo et al also used a combination

of Fuzzy logic and Genetic Algorithms for the

heading control of another kind of autonomous

underwater robot [7] One of the most popular

intelligent controllers used for motion control of

fish robots is the Central Pattern Generator

(CPG) This CPG controller was used by Long

Wang et al [8], Daisy Lachat et al [9], and Wei

Zhao et al [10] for their fish robots Another type

of controller, called a hybrid controller, is also

used in the motion control of fish robots This

type of controller is proposed by Jindong Liu et

al [11] However, all the studies discussed above

are based on simplified dynamic models or

experiments involving fish robots Besides, there

are not many applications of Sliding Mode

Controllers used in fish robot Most uses of

Sliding Mode Controllers are in the fields of

other robotics [12, 13], mechanical system [14,

15], or motor control [16, 17]

In this paper, we considered a 3-joint (4-link)

Carangiform fish robot type The dynamic model

of this robot was derived using the Lagrange

method The influences of fluid force on the

motion of the fish robot are also considered,

based on M J Lighthill’s Carangiform

Decomposition (SVD) algorithm is also used in

our simulation program to minimize the

divergence of the fish robot’s links when

environment The dynamic model of the fish

robot in this paper is analyzed, including the

heading angle’s motion of the fish robot This

concept differs from the kinematic equation

proposed by M J Lighthill [18], in which that the body-spline takes the form of a traveling wave With this kind of dynamic equation, we can analyze more precisely the turning and heading angles of fish robots when considering their operation in underwater environment The main goal of this paper is the introduction

of a new dynamic analysis concept Normally, the head and body of the Carangiform fish robot

is supposed to be rigid, and these undulate as they swim However, there is no method to express the heading angle of a fish robot at each sampling time of operation Therefore, in our dynamic analysis approach, we consider the heading angle of the fish robot’s head-body part With this method, we can easily recognize the heading angle of the fish robot at each sampling time during operation, which is also helpful when researching the turning motion of the fish robot The second point of this paper is that we propose

a SMC and a FSMC controllers to design the straight motion and turning motion controllers for

a fish robot The FSMC provides excellent performance in both straight and turning control

of a fish robot in comparison to the SMC for fish robot

2 DYNAMICS ANALYSIS

Increasing size of movement

Posterior part Anterior part

Caudal part

Tail fin Main axis

Transverse axis Pectoral fin

Figure 1 Carangiform locomotion style

In our fish robot, we focus mainly on the Carangiform fish type because of its fast

mackerel or trout The Carangiform fish type has

a large tail with a high aspect ratio The movement of this fish requires powerful muscles that generate side-to-side motion in the posterior part Also, the anterior part of the fish robot undulates while operating, as shown in Fig 1 above

We design a 3-joint (4 links) fish robot in order to get smoother and more natural motion The analytical model of the fish robot is shown in Fig 2 In this figure, the head and body of the

Trang 16

a0

m0 (x0,y0)

T2

m3 (x3,y3)

m2 (x2,y2)

a3

l3

a2 l2

l1

a1

X

Y

m1 (x1,y1)

T1

(link1)

(link2)

(link3)

(link0)

l0

Figure 2 Fish robot analytical model

joint 2, respectively, which are generated by two

active DC motors We assume that the inertial

fluid force, FV, and the lift force, FJ, act on the

tail fin only (link 3), which is similar to the concept of Motomu Nakashima et al [19] and is explained in our previous research [20]

FC

V

F

J

F

F

F

FD Direction of movement

X Y

Figure 3 Forces distribution on the fish robot

The force distribution on the fish robot is

component at the tail fin, FC is the lateral force

from the motion of the fish robot The calculation

of these forces, including F F F F F , and V, J, C, F, D

previous research [20]

By using Lagrange’s method, the dynamic

model of the fish robot is described briefly by Eq

(1)

q q q q

é ù

ê ú

ê ú

&

&

&

&

(1)

By solving Eq (1), we can determine the values of q i and q&i (i = 0  3) The motion equation of the fish robot is expressed in Eq (2)

G

x & & is the acceleration of the fish robot’s

force caused by the friction between the fish

robot and the surrounding environment when the

fish robot swims

2

1

2

F = r V C S (3)

velocity of the fish robot relative to the water

area of the main body of the fish robot, which is projected on the perpendicular plane of the flow

The values of all parameters in Eqs (2-3) are referred to in our previous research [20]

3 SLIDING MODE BASED CONTROLLERS

In this section, a SMC and a FSMC are proposed to make a fish robot to follow a straight path with a

predefined heading angle or to turn toward a heading direction with a desired turning angle

3.1 The Sliding Mode Controller Design

The SMC system for heading and turning control of the fish robot is introduced in Fig 4

Desired heading 

y

d/dt

e

SMC

Fish robot (G)

de

u

 dt

D

Figure 4 SMC controller system

In the heading and turning control of a fish

robot, we consider only the yaw angle of the fish

robot

G= q&+ - q& D: Disturbance of the

surround environment We assume that the

uG D

From Eq (4) and Eq (5), we have:

e b uG D

We then calculate the average error during the

relevant time:

0

1 t

t

0

dt l dt l t

s& is calculated, as follow:

The sliding mode control input is described by:

( )

s

u =h e sign s& (11)

( )

1

max

eq

u G- b D sign s l e

= &+ + (12)

( )

1

max

u G- b D sign s l e h e sign s

(13)

To prove the convergence of the sliding mode, we consider the derivative of the distance

Trang 18

2

0.5

have:

V&= ss& (14)

s

s

V D D sign s G e sign s

the system is asymptotically stable Therefore,

can be applied to the heading and turning control

of the fish robot

3.2 Fuzzy Sliding Mode Controller Design

G1 Desired heading 

y

G2

1

e

FSMC du

G3 Fish

robot

s ds

u

Figure 5 FSMC controller system

However, the disadvantage of the SMC is that

the discontinuous in the control signal causes

chattering There are many methods used to

reduce the chattering phenomenon, like changing

the saturation function In our fish robot, we use

the combination of Sliding Mode Control and

Fuzzy Logic Control (FLC) to design the

direction controller for fish robot Based on the

calculate the suitable value of the control signal

above The equations of the FSMC are presented

briefly in Eqs (17 – 22)

1

( ) * 1( ) 2( )

s k = c e k +e k (19)

( ) ( ) ( 1)

du k = FLC s k ds kéê ùú

For the FLC, the number linguistic terms for

each linguistic variable are three for two inputs

and five for one output The three linguistic

(Zero) and P (Positive) The five linguistic

(Negative Small), ZE (Zero), PS (Positive Small)

and PB (Positive Big) The triangle-type

membership function is chosen for the system The center of gravity method is chosen as the defuzzification method A total of nine rules are applied to the Fuzzy controller

4 SIMULATION RESULTS

For simulation, we consider that the total length of the fish robot is about 450 mm, including 3 links and the tail fin Two external input torques are applied to joint1 and joint2 of the fish robot to generate propulsion The head and body of fish robot are supposed to be one rigid part (link0) which is connected to link1 by active RC motor1 (joint1) Then, link1 and link2 are connected by active RC motor2 (joint2) Lastly, link3 (lunate shape tail fin) is jointed into link2 (joint3) by two extension flexible springs in order to imitate the smooth motion of real fish The stiffness value of each spring is about 100Nm Total weight of the fish robot (in air) is about 5 kg Simulations are performed to evaluate the tracking performance to follow straight paths with a heading angle and angular paths with a turning angle The desired heading angles for the fish robot are selected as 30 degrees and 60 degrees, and the same angles are selected for the case of the turning angle

In the simulation, we consider two kinds of input disturbances for the flow velocity to check the robustness of the controllers The first is the

(23)[21] The disturbance impacts the fish robot

at every sampling time during the entire

operation time We assume that the velocity of

previous research [20], and we also limit the

range of the continuous disturbance such as

[ 0.25, 0.25]( / )

c

sudden disturbance also impacts the flow velocity

at some arbitrary sampling times while the fish

( ) ( )

2*log sin

c

s

w = R whereR Î [ ]0,1. (24)

(25)

0.08

4.1 Tracking Control along a Straight Path

Joint1

Joint2

Joint3



CCW

CW



Joint3 Joint2

Joint1

Figure 6 Turning motion of a fish robot in counterclockwise (CCW) and clockwise (CW) directions

In both straight motion mode and turning

motion mode, the direction control of the fish

robot is necessary to recover the tracking error

The change of direction can be achieved by

oscillating each link that is operated by the

corresponding input torque at each joint Fig 6

shows examples of direction changes of the fish

robot for the CW or CCW direction

4.1.1 Tracking Control using the SMC

The principle of the SMC is introduced in Fig

4, and the control signal is presented as Eq (13)

performance of the fish robot, in which the head

of the robot should follow a straight path with a

that the fish robot follows the path with an error less than 1 degree, and the sum square error during the whole period of operation (60

flow velocity with the disturbance that affects the original SMC control system

a b

Figure 7 a Direction control result by SMC (desired heading angle=30 degrees)

b Applied flow disturbance

0 10 20 30 40 50 60

29.4

29.6

29.8

30

30.2

30.4

30.6

30.8

Direction control of fish robot in straight motion - SMC controller

Time (s) -0.10 10 20 30 40 50 60

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Applied flow disturbance

Time (s)

Trang 20

a b

Figure 8 a Direction control result by SMC (desired heading angle=60 degrees)

b Applied flow disturbance

Fig 8 shows the direction control result for

the heading angle of 60 degrees, in which the fish

robot follows the path with an error also less than

1 degree, and the sum square error during the

simulations exemplify that the SMC provides

quite robustness, as well as satisfactory tracking

performance, even in the flow disturbance

environment

4.1.2 Tracking Control using the FSMC

The principle of the FSMC is introduced in

Fig 5 This section discusses the application of

FSMC to controlling the heading motion for fish robot The testing values of desired heading angle

or desired yaw angle are also selected of 30 degrees and 60 degrees, respectively The results are introduced in Fig 9 and Fig 10 These figures describe the performance of fish robot’s motion when applying the FSMC to the heading control These figures show that, even though the influences of flow disturbances are also considered, the motion of the fish robot is quite good and stable

a b

Figure 9 a Direction control result by FSMC (desired heading angle=30 degrees)

b Applied flow disturbance

0 10 20 30 40 50 60

59.6

59.8

60

60.2

60.4

60.6

60.8

61

Direction control of fish robot in straight motion - SMC controller

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Applied flow disturbance

Time (s)

0 10 20 30 40 50 60

29.4

29.6

29.8

30

30.2

30.4

30.6

30.8

Direction control of fish robot in straight motion - FSMC controller

Time (s)

0 10 20 30 40 50 60 -0.15

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Applied flow disturbance

Time (s)

a b

Figure 10 a Direction control result by FSMC (desired heading angle=60 degrees)

b Applied flow disturbance

The sum square errors when using this

desired heading angle equal to both 30 degrees

and 60 degrees From these results, the

performances of the fish robot’s heading angle

are better when testing with the FSMC than when

using the SMC The sum square errors when

using FSMC are also smaller than when using the

SMC Therefore, the FSMC is better than the

SMC in controlling the straight motion of the fish

robot From the performances of fish robot in the

figures above, the SMC and FSMC are quite

robust controllers in the heading control problem

of the fish robot

4.2 Tracking Control for Turning Motion

In this turning mode, the controller controls turns of the fish robot with the desired turning angle After the fish robot reaches the desired turning angle, it swims straight with the desired heading angle or desired yaw angle, which is equal to the value of the turning angle The desired turning angles to test the controllers are

30 degrees and 60 degrees The fish robot is controlled to start turning from 0 degree to the desired turning angle

4.2.1 Turning Control using the SMC

a b

Figure 11 a Turning control performance of the SMC (desired turning angle=300)

b Applied flow disturbance

0 10 20 30 40 50 60

59.4

59.6

59.8

60

60.2

60.4

60.6

60.8

Direction control of fish robot in straight motion - FSMC controller

Time (s)

0 10 20 30 40 50 60 -0.1

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Applied flow disturbance

Time (s)

0 10 20 30 40 50 60

0

5

10

15

20

25

30

35

Direction control of fish robot in turn & straight motion - SMC controller

Time (s)

0 10 20 30 40 50 60 -0.1

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Applied flow disturbance

Time (s)

Trang 22

a b

Figure 12 a Turning control performance of the SMC (desired turning angle=600)

b Applied flow disturbance

The principle of the SMC is exactly the same

as the case of straight motion expressed in Fig 4

The only difference is that the desired heading

angle, b , is changed to the desired turning angle,

b The simulation results for turning angles of

30 degrees and 60 degrees are shown in Figs 11

and 12 The fish robot performs quite well with

the SMC

The time required for the SMC to turn the fish robot to the desired turning angles are about 1.9 seconds for turning of 30 degrees and about 2.3 seconds for 60 degrees The steady state errors in

4.2.2 Turning Control using the FSMC

a b

Figure 13 a Turning control performance of the FSMC (desired turning angle=300)

b Applied flow disturbance

The testing of the FSMC in turning motion is

also conducted similar to that for the previous

controller The desired heading angle, b , in Fig

5 is substituted by the desired turning angle, b

The performance of the fish robot in turning

mode of 30 degrees and 60 degrees are presented

in Figs 13 and 14, respectively The time required for the FSMC to turn the fish robot to the desire turning angle is about 1.82 seconds and 2.2 seconds for 30 degrees and 60 degrees,

0 10 20 30 40 50 60

0

10

20

30

40

50

60

70

Direction control of fish robot in turn & straight motion - SMC controller

Time (s)

0 10 20 30 40 50 60 -0.15

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Applied flow disturbance

Time (s)

0 10 20 30 40 50 60

0

5

10

15

20

25

30

35

Direction control of fish robot in turn & straight motion - FSMC controller

Time (s)

0 10 20 30 40 50 60 -0.1

-0.05 0 0.05 0.1 0.15 0.2

0.25

Noise of surround environment

Time (s)

respectively The steady state error for these cases is 0.230 and 0.220, respectively

a b

Figure 14 a Turning control performance of the FSMC (desired turning angle=600)

b Applied flow disturbance

When applying the FSMC in this motion,

the fish robot also performs a little better than

when applying the SMC Also, the error of the

fish robot is quite small Therefore, the SMC

and FSMC are good controllers for the turning

motion of the fish robot

5 CONCLUSION

This paper presents a model of a 3-joint

Carangiform fish robot From this type of fish

robot, a new dynamic model is derived using

Lagrange’s method This type of dynamic also

includes the motion of the head and body of the

fish robot, a characteristic difference between

this dynamic analysis and other conventional

analyses of the dynamics of Carangiform fish

robots The influence of the fluid forces exerted

on the motion of the fish robot in underwater

environment is also considered in the dynamic

model by using the concept of M J Lighthill’s

Carangiform propulsion Moreover, the SVD algorithm is also used in our simulation program as an effective method to reduce the divergence of the fish robot links’ movement when solving the matrix of the dynamic model

In this paper, the SMC and FSMC are also good for turning motion control for fish robot Besides, both the SMC and FSMC are quite simple controllers, but they are highly effective

in controlling motion problems for the fish robot Besides, some experiments will be carried out in the near future to check the agreement between the simulation results and the experimental results

ACKNOWLEDGEMENT

This research is funded by Viet Nam National University

Ho Chi Minh City (VNU-HCM) under Grant number B-2013-20-01

0 10 20 30 40 50 60

0

10

20

30

40

50

60

70

Direction control of fish robot in turn & straight motion - FSMC controller

Time (s)

0 10 20 30 40 50 60 -0.15

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Noise of surround environment

Time (s)